Multivariable Calculus Online

The Structure of the Textbook 

Although the day-to-day experiences of students are similar to that of traditional courses, Multivariable Calculus Online differs from a traditional multivariable textbook in many ways.  This differences are due both to the desire for meaningful uses of technology and the increased need for coherence and continuity in the organization of the course.

For example, quadric surfaces usually occur near the beginning of a traditional textbook, and the emphasis is usually on what types of quadratic equations lead to which types of surfaces.  In our textbook, quadric surfaces are placed in the third chapter, where they are introduced as level surfaces that are related to certain parametric surfaces. This placement allows more meaningful uses of technology than would have been possible had quadric surfaces been placed any earlier.

We have found that relocating elementary topics like quadric surfaces, polar coordinates, and cylindrical coordinates so that they occur after the introduction of the partial derivative allows them to be discussed more substantively.  This can be seen in the contents of the textbook, which is divided into five chapters with an optional “capstone”:

1. Vector-Valued Functions:  Vectors, dot product, cross product, parameterized curves, velocity, speed, acceleration, arclength, unit tangent vector, unit normal vector, and curvature.
2. Functions of 2 Variables:  Graphs, domains, limits, partial derivatives, partial differential equations, differentiability, tangent planes, the chain rule, gradients and level curves, directional derivatives, optimization, and Lagrange Multipliers.
3. Surfaces and Transformations: Level surfaces, tangent planes, parametric surfaces, coordinate transformations, the Jacobian, polar coordinates, cylindrical coordinates, spherical coordinates, the unit surface normal, geodesics, the fundamental form, and curvature of a surface.
4. Multiple Integrals:  Iterated integrals, the double integral, applications, change of variable, integration in polar coordinates, triple integrals, integration in spherical and cylindrical coordinates.
5. Fundamental Theorems:  Vector fields, divergence, curl, line integrals, potentials, fundamental theorem for line integrals, Green’s theorem, the Divergence theorem, Stoke’s theorem, Stoke’s theorem for differential forms.

Chapters 1, 2, 4, and 5 are familiar to anyone who has taught multivariable calculus, but chapter 3 is different for the reason mentioned above.  Moreover, chapter 3 also contains a discussion of coordinate transformations and the Jacobian, thus preparing students for the theorem in chapter 4 on the change of variables in multiple integrals. 

There are also many new topics, as well as new applications and new insights into familiar topics.  However, our experience is that the pace of the course is similar to that of a traditional course, in that most instructors will be able to finish the text or at least get to Green’s theorem without the course seeming to be hurried or overly demanding.